Lord Kelvin Part 5

In the earlier years of his professorship Professor Thomson taught his cla.s.s entirely himself, and gathered round him, as he has told us in the Bangor address, an enthusiastic band of workers who aided him in the researches which he began on the electrodynamic qualities of metals, the elastic properties of substances, the thermal and electrical conductivities of metals, and at a later date in the electric and magnetic work which he undertook as a member of the British a.s.sociation Committee on Electrical Standards. The cla.s.s met, as has been stated, twice a day, first for lectures, then for exercises and oral examination. The changes which took place later in the curriculum, and especially the introduction of honours cla.s.ses in the different subjects, rendered it difficult, if not impossible, for two hours'

attendance to be given daily on all subjects, and students were at first excused attendance at the second hour, and finally such attendance became practically optional. But so long as the old traditional curriculum in Arts--of Humanity, Greek, Logic, Mathematics, Moral Philosophy and Natural Philosophy--endured, a large number of students found it profitable to attend at both hours, and it was possible to give a large amount of excellent tutorial instruction by the working of examples and oral examination.

Thomson always held that his commission included the subject of physical astronomy, and though his lectures on that subject were, as a rule, confined to a statement of Kepler's laws and Newton's deductions from them, he took care that the written and oral examinations included astronomical questions, for which the students were enjoined to prepare by reading Herschel's Outlines, or some similar text-book. This injunction not infrequently was disregarded, and discomfiture of the student followed as a matter of course, if he was called on to answer.

Nor were the questions always easy to prepare for by reading. A man might have a fair knowledge of elementary astronomy, and be unable to answer offhand such a question as, "Why is the ecliptic called the ecliptic?" or to say, when the lectures on Kepler had been omitted, short and tersely just what was Newton's deduction from the third law of the planetary motions.

Home exercises were not prescribed as part of the regular work except from time to time in the "Higher Mathematical Cla.s.s" which for thirty years or more of Thomson's tenure of office was held in the department.

But the whole ordinary cla.s.s met every Monday morning and spent the usual lecture hour in answering a paper of dynamical and physical questions. As many as ten, and sometimes eleven, questions were set in these papers, some of them fairly difficult and involving novel ideas, and by this weekly paper of problems the best students, a dozen or more perhaps, were helped to acquire a faculty of prompt and brief expression. It was not uncommon for a good man to score 80 or 90 or even 100 per cent. in the paper, no small feat to accomplish in a single hour. But to a considerable majority of the cla.s.s, it is doubtful whether the weekly examination was of much advantage: they attempted one or two of the more descriptive questions perhaps, but a good many did next to nothing. The examinations came every week, and so the preparation for one after another was neglected, and as much procrastination of work ensued as there would have been if only four or five papers a session had been prescribed. Then the work of looking over so many papers was a heavy task to the professor's a.s.sistant, a task which became impossible when, for a few years in the early 'eighties, the students in the ordinary cla.s.s numbered about 250.

The subject of natural philosophy had become so extensive in 1846 that Professor J. P. Nichol called attention to the necessity for special arrangements for its adequate teaching. What would he say if he could survey its dimensions at the present time! To give even a brief outline of the princ.i.p.al topics in dynamics, heat, acoustics, light, magnetism, and electricity is more than can be accomplished in any course of university lectures; and the only way to teach well and economically the large numbers of students[16] who now throng the physics cla.s.ses is to give each week, say, three lectures as well considered and arranged as possible, without any interruption from oral examination, and a.s.semble the students in smaller cla.s.ses two or three times a week for exercises and oral examination.

Thomson stated his views as to examinations and lectures in the Bangor address. "The object of a university is teaching, not testing, ... in respect to the teaching of a university the object of examination is to promote the teaching. The examination should be, in the first place, daily. No professor should meet his cla.s.s without talking to them. He should talk to them and they to him. The French call a lecture a conference, and I admire that idea. Every lecture should be a conference of teachers and students. It is the true ideal of a professorial lecture. I have found that many students are afflicted when they come up to college with the disease called 'aphasia.' They will not answer when questioned, even when the very words of the answer are put in their mouths, or when the answer is simply 'yes' or 'no.' That disease wears off in a few weeks, but the great cure for it is in repeated and careful and very free interchange of question and answer between teacher and student.... Written examinations are very important, as training the student to express with clearness and accuracy the knowledge he has gained, but they should be once a week to be beneficial."

The great difficulty now, when both cla.s.ses and subject have grown enormously, is to have free conversation between professor and student, and yet give an adequate account of the subject. To examine orally in a thorough way two students in each cla.s.s-hour is about as much as can be done if there is to be any systematic exposition by lecture at all; and thus the conference between teacher and individual student can occur only twice a year at most. Nevertheless Lord Kelvin was undoubtedly right: oral examination and the training of individual students in the art of clear and ready expression are very desirable. The real difficulties of the subject are those which occur to the best students, and a discussion of them in the presence of others is good for all. This is difficult nowadays, for large cla.s.ses cannot afford to wait while two or three backward students grope after answers to questions--which in many cases must be on points which are sufficiently plain to the majority--to say nothing of the temptation to disorder which the display of personal peculiarities or oddities of expression generally affords to an a.s.sembly of students. But time will be economised and many advantages added, if large cla.s.ses are split up into sections for tutorial work, to supplement the careful presentation of the subject made in the systematic lectures delivered to the whole cla.s.s in each case. The introduction of a tutorial system will, however, do far more harm than good, unless the method of instruction is such as to foster the self-reliance of the student, who must not be, so to speak, spoon-fed: such a method, and the advantages of the weekly examination on paper may be secured, by setting the tutorial cla.s.s to work out on the spot exercises prescribed by the lecturer. But the danger, which is a very real one, can only be fully avoided by the precautions of a skilful teacher, who in those small cla.s.ses will draw out and direct the ideas of his students, rather than impart knowledge directly.

After a few years Thomson found it necessary to appoint an a.s.sistant, and Mr. Donald McFarlane, who had distinguished himself in the Mathematics and Natural Philosophy cla.s.ses, was chosen. Mr. McFarlane was originally a block-printer, and seems to have been an apprentice at Alexandria in the Vale of Leven, at the time of the pa.s.sing of the first Reform Bill. After some time spent in the cotton industry of the district, he became a teacher in a village school in the Vale of Leven, and afterwards entered the University as a student. He discharged his duties in the most faithful and self-abnegating manner until his retirement in 1880, when he had become advanced in years. He had charge of the instruments of the department, got ready the lecture ill.u.s.trations and attended during lecture to a.s.sist in the experiments and supply numerical data when required, prepared the weekly cla.s.s examination paper and read the answers handed in, and a.s.sisted in the original investigations which the professor was always enthusiastically pursuing. A kind of universal physical genius was McFarlane; an expert calculator and an exact and careful experimentalist. Many a long and involved arithmetical research he carried out, much apparatus he made in a homely way, and much he repaired and adjusted. Then, always when the professor was out of the way and calm had descended on the apparatus-room, if not on the laboratory, McFarlane sat down to reduce his pile of examination papers, lest Monday should arrive with a new deluge of crude answers and queer mistakes, ere the former had disappeared. On Friday afternoons at 3 o'clock he gave solutions of the previous Monday's questions to any members of the cla.s.s who cared to attend; and his clear and deliberate explanations were much appreciated.

An unfailing tribute was rendered to him every year by the students, and often took the form of a valuable gift for which one and all had subscribed. A recluse he was in his way, hardly anybody knew where he lived--the professor certainly did not--and a man of the highest ability and of the most absolute unselfishness. An hour in the evening with one or two special friends, and the study of German, were the only recreations of McFarlane's solitary life. He was full of humour, and told with keen enjoyment stories of the University worthies of a bygone age. For thirty years he worked on for a meagre salary, for during the earlier part of that time no provision for a.s.sistants was made in the Government grant to the Scottish Universities. By an ordinance issued in 1861 by the University Commissioners, appointed under the Act of 1858, a grant of 100 a year was made from the Consolidated Fund for an a.s.sistant in each of the departments of Humanity, Greek, Mathematics, and Natural Philosophy, and for two in the department of Chemistry; and McFarlane's position was somewhat improved. His veneration for Thomson was such as few students or a.s.sistants have had for a master: his devotion resembled that of the old famulus rather than the much more measured respect paid by modern a.s.sistants to their chiefs.

After his retirement McFarlane lived on in Glasgow, and amused himself reading out-of-the-way Latin literature and with the calculation of eclipses! He finally returned to Alexandria, where he died in February 1897. "Old McFarlane" will be held in affectionate remembrance so long as students of the Natural Philosophy Cla.s.s in the 'fifties and 'sixties and 'seventies, now, alas! a fast vanishing band, survive.

Soon after taking his degree of B.A. at Cambridge in 1845, Thomson had been elected a Fellow of St. Peter's College. In 1852 he vacated his Fellowship on his marriage to Miss Margaret Crum, daughter of Mr. Walter Crum of Thornliebank, near Glasgow, but was re-elected in 1871, and remained thereafter a Fellow of Peterhouse throughout his life.

CHAPTER VII

THE "ACCOUNT OF CARNOT'S THEORY OF THE MOTIVE POWER OF HEAT"--TRANSITION TO THE DYNAMICAL THEORY OF HEAT

The meeting of Thomson and Joule at Oxford in 1847 was fraught with important results to the theory of heat. Thomson had previously become acquainted with Carnot's essay, most probably through Clapeyron's account of it in the _Journal de l'ecole Polytechnique_, 1834, and had adopted Carnot's view that when work was done by a heat engine heat was merely let down from a body at one temperature to a body at a lower temperature. Joule apparently knew nothing of Carnot's theory, and had therefore come to the consideration of the subject without any preconceived opinions. He had thus been led to form a clear notion of heat as something which could be transformed into work, and _vice versa_. This was the root idea of his attempt to find the dynamical equivalent of heat. It was obvious that a heat engine took heat from a source and gave heat to a refrigerator, and Joule naturally concluded that the appearance of the work done by the engine must be accompanied by the disappearance of a quant.i.ty of heat of which the work done was the equivalent. He carried this idea consistently through all his work upon energy-changes, not merely in heat engines but in what might be called electric engines. For he pointed out that the heat produced in the circuit of a voltaic battery was the equivalent of the energy-changes within the battery, and that, moreover, when an electromagnetic engine was driven by the current, or when electrochemical decomposition was effected in a voltameter in the circuit, the heat evolved in the circuit for a given expenditure of the materials of the battery was less than it would otherwise have been, by the equivalent of the work done by the engine, or of the chemical changes effected in the voltameter. Thus Joule was in possession at an earlier date than Thomson of the fundamental notion upon which the true dynamical theory of heat engines is founded. Thomson, on the other hand, as soon as he had received this idea, was able to add to it the conception, derived from Carnot, of a reversible engine as the engine of greatest efficiency, and to deduce in a highly original manner all the consequences of these doctrines which go to make up the ordinary thermodynamics even of the present time. Though Clausius was the first, as we shall see, to deduce various important theorems, yet Thomson's discussion of the question had a quality peculiarly its own. It was marked by that freedom from unstated a.s.sumptions, from extraneous considerations, from vagueness of statement and of thought, which characterises all his applications of mathematics to physics. The physical ideas are always set forth clearly and in such a manner that their quant.i.tative representation is immediate: we shall have an example of this in the doctrine of absolute temperature. In most of the thermodynamical discussions which take the great memoir of Clausius as their starting point, temperature is supposed to be given by a hypothetical something which is called a perfect gas, and it is very difficult, if not impossible, to gather a precise notion of the properties of such a gas and of the temperature scale thereon founded.

Thomson's scale enables a perfect gas to be defined, and the deviations of the properties of ordinary gases from those of such a gas to be observed and measured.

The idea, then, which Joule had communicated to Section A, when Thomson interposed to call attention to its importance, was that work spent in overcoming friction had its equivalent in the heat produced, that, in fact, the amount of heat generated in such a case was proportional to the work spent, quite irrespective of the materials used in the process, provided no change of the internal energy of any of them took place so as to affect the resulting quant.i.ty of heat. This forced upon physicists the view pointed to by the doctrine of the immateriality of heat, established by the experiments of Rumford and Davy, that heat itself was a form of energy; and thus the principle of conservation of energy was freed from its one defect, its apparent failure when work was done against friction.

Rumford had noted the very great evolution of heat when gun-metal was rubbed by a blunt borer, and had come to the reasonable conclusion that what was evolved in apparently unlimited quant.i.ty by the abrasion or cutting down of a negligible quant.i.ty of materials could not be a material substance. He had also made a rough estimate of the relation between the work spent in driving the borer by horse-power and the heat generated. Joule's method of determining the work-equivalent of heat was a refinement of Rumford's, but differed in the all-important respect that accurate means were employed for measuring the expenditure of work and the gain of heat. He stirred a liquid, such as water or mercury, in a kind of churn driven by a falling weight. The range of descent of the weight enabled the work consumed to be exactly estimated, and a sensitive thermometer in the liquid measured the rise of temperature; thus the heat produced was accurately determined. The rise of temperature was very slight, and the change of state of the liquid, and therefore any possible change in its internal energy, was infinitesimal.

The experiments were carried out with great care, and included very exact measurements of the various corrections--for example, the amount of work spent at pulleys and pivots without affecting the liquid, and the loss of heat by radiation. The experiments proved that the work spent on the liquid and the heat produced were in direct proportion to one another. He found, finally, in 1850, that 772 foot-pounds of work at Manchester generated one British thermal unit, that is, as much heat as sufficed to raise a pound of water from 60 F. to 61 F. An approximation to this conclusion was contained in the paper which he communicated to the British a.s.sociation at Oxford in 1847.

The results of a later determination made with an improved apparatus, and completed in 1878, gave a very slightly higher result. When corrected to the corresponding Fahrenheit degree on the air thermometer it must be increased by somewhat less than one per cent. The exact relation has been the subject during the last twenty years of much refined experimental work, but without any serious alteration of the number indicated above.

It is probable that in consequence of the conference which he had with Joule at Oxford Thomson had his thoughts turned for some time almost exclusively to the dynamical theory of heat engines. He worked at the subject almost continuously for a long time, sending paper after paper to the Edinburgh Royal Society. As we have seen, he had given Joule a description of Carnot's essay on the Motive Power of Heat and the conclusions, or some of them, therein contained. Joule's result, and the thermodynamic law which it established, gave the key to the correction of Carnot's theory necessary to bring it into line with a complete doctrine of energy, which should take account of work done against frictional resistances.

Mayer of Heilbronn had endeavoured to determine the dynamical equivalent of heat in 1842, by calculating from the knowledge available at the time of the two specific heats of air--the specific heat at constant pressure and the specific heat at constant volume--the heat value of the work spent in compressing air from a given volume to a smaller one. The principle of this determination is easily understood, but it involves an a.s.sumption that is not always clearly perceived. Let the air be imagined confined in a cylinder closed by a frictionless piston, which is kept from moving out under the air pressure by force applied from without.

Let heat be given to the air so as to raise its temperature, while the piston moves out so as to keep the pressure constant. If the pressure be p and the increase of volume be dv, the work done against the external force is pdv. Let the rise of temperature be one degree of the Centigrade scale, and the ma.s.s of air be one gramme, the heat given to the gas is the specific heat Cp of the gas at constant pressure, for there is only slight variation of specific heat with temperature. But if the piston had been fixed the heat required for the same rise of temperature would have been Cv, the specific heat at constant volume.

Now Mayer a.s.sumed that the excess of the specific heat Cp above Cv was the thermal equivalent of the work pdv done in the former case. Thus he obtained the equation J(Cp - Cv) = pdv, where J denotes the dynamical equivalent of heat and Cp, Cv are taken in thermal units. But if a be the coefficient of expansion of the air under constant pressure (that is 1?273), and v0 be the volume of the air at 0 C., we have dv = av0, so that J(Cp - Cv) = apv0. Now if p be one atmosphere, say 1.014 10^6 dynes per square centimetre, and the temperature be the freezing point of water, the volume of a gramme of air is 1?.001293 in cubic centimetres. Hence

J(Cp - Cv) = (1.014 10^6)?(273 .001293)

from which, if Cp - Cv is known, the value of J can be found.

In Mayer's time the difference of the specific heats of air was imperfectly known, and so J could not be found with anything like accuracy. From Regnault's experiments on the specific heat at constant pressure, and from the known ratio of the specific heats as deduced from the velocity of sound combined with Regnault's result, the value of Cp - Cv may be taken as .0686. Thus J works out to 42.2 10^6, in ergs per calorie, which is not far from the true value. Mayer obtained a result equivalent to 36.5 10^6 ergs per calorie.

The a.s.sumption on which this calculation is founded is that there is no alteration of the internal energy of the gas in consequence of expansion. If the air when raised in temperature, and at the same time increased in volume, contained less internal energy than when simply heated without alteration of volume, the energy evolved would be available to aid the performance of the work done against external forces, and less heat would be required, or, in the contrary case, more heat would be required, than would be necessary if the internal energy remained unaltered. Thus putting dW for pdv, the work done, e for the internal energy before expansion, and dH for the heat given to the gas, we have obviously the equation

JdH = de + dW

where de is the change of internal energy due to the alteration of volume, together with the alteration of temperature. If now the temperature be altered without expansion, no external work is done and dW for that case is zero. Let ?e and ?H be the energy change and the heat supplied, then in this case

J?H = ?e + O

Thus

J(dH - ?H) = de - ?e + dW

and the a.s.sumption is that de = ?e, so that dW = J(dH - ?H); that is, dW = J(Cp - Cv), when the rise of temperature is 1 C. and the ma.s.s of air is one gramme. This a.s.sumption requires justification, and by an experiment of Joule's, which was repeated in a more sensitive form devised by Thomson, it was shown to be a very close approximation to the truth. Joule's experiment is well known: the explanation given above may serve to make clear the nature of the research undertaken later by Thomson and Joule conjointly.

The inverse process, the conversion of heat into work, required investigation, and it is this that const.i.tutes the science of thermodynamics. It was the subject of the celebrated _Reflexions sur la Puissance Motrice du Feu, et sur les Machines Propres a Developper cette Puissance_, published in 1824 by Sadi Carnot, an uncle of the late President of the French Republic. Only a few copies of this essay were issued, and its text was known to very few persons twenty-four years later, when it was reprinted by the Academy of Sciences. Its methods and conclusions were set forth by Thomson in 1849 in a memoir which he ent.i.tled, "An Account of Carnot's Theory of the Motive Power of Heat."

Numerical results deduced from Regnault's experiments on steam were included; and the memoir as a whole led naturally in Thomson's hands to a corrected theory of heat engines, which he published in 1852. Carnot's view of the working of a heat engine was founded on the a.n.a.logy of the performance of work by a stream of water descending from a higher level to a lower. The same quant.i.ty of water flows away in a given time from a water wheel in the tail-race as is received in that time by the wheel from the supply stream. Now a heat engine receives heat from a supplying body, or source, at one temperature and parts with heat to another body (for example, the condenser of a steam engine) at a lower temperature.

This body is usually called the refrigerator. According to Carnot these temperatures corresponded to the two levels in the case of the water wheel; the heat was what flowed through the engine. Thus in his theory as much heat was given up by a heat engine to the body at the lower temperature as was received by it from the source. The heat was simply transferred from the body at the higher temperature to the body at the lower; and this transference was supposed to be the source of the work.[17]

The first law of thermodynamics based on Joule's proportionality of heat produced to work expended, and the converse a.s.sumed and verified _a posteriori_, showed that this view is erroneous, and that the heat delivered to the refrigerator must be less in amount than that received from the source, by exactly the amount which is converted into work, together with the heat which, in an imperfect engine, is lost by conduction, etc., from the cylinder or other working chamber. This change was made by Thomson in his second paper: but he found the ideas of Carnot of direct and fruitful application in the new theory. These were the cycle of operations and the ideal reversible engine.

In the Carnot cycle the working substance--which might be a gas or a vapour, or a liquid, or a vapour and its liquid in contact: it did not matter what for the result--was supposed to be put through a succession of changes in which the final state coincided with the initial. Thus the substance having been brought back to the same physical condition as it had when the cycle began, has the same internal energy as it had at the beginning, and in the reckoning of the work done by or against external forces, nothing requires to be set to the account of the working substance. This is the first great advantage of the method of reasoning which Carnot introduced.

The ideal engine was a very simple affair: but the notion of reversibility is difficult to express in a form sufficiently definite and precise. Carnot does not attempt this; he merely contents himself with describing certain cycles of operations which obviously can be carried through in the reverse order. Nor does Thomson go further in his "Account of Carnot's Theory," though he states the criterion of a perfect engine in the words, "A perfect thermodynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency, if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced." This proposition was proved by Carnot: and the following formal statement in the essay is made: "La puissance motrice de la chaleur est independante des agents mis en uvre pour la realiser: sa quant.i.te est fixee uniquement par les temperatures des corps entre lesquels se fait, en dernier resultat, le transport du calorique." The result involved in each, that the work done in a cycle by an ideal engine depends on the temperatures between which it works and not at all on the working substance, is, as we shall see, of the greatest importance. The proof of the proposition, by supposing a more efficient engine than the ideal one to exist, and to be coupled with the latter, so that the more efficient would perform the cycle forwards and the ideal engine the same cycle backwards, is well known. In Carnot's view the former would do more work by letting down a given quant.i.ty of heat from the higher to the lower temperature than was spent on the latter in transferring the same quant.i.ty of heat from the lower to the higher temperature, so that no heat would be taken from or given to source or refrigerator, while there would be a gain of work on the whole. This would be equivalent to admitting that useful work could be continually performed without any resulting thermal or other change in the agents performing the work. Even at that time this could not be admitted as possible, and hence the supposition that a more efficient engine than the reversible one could exist was untenable.

Carnot showed that the work done by an ideal engine, in transferring heat from one temperature to another, was to be found by means of a certain function of the temperature, hence called "Carnot's function."

The corresponding function in the true dynamical theory is always called Carnot's. A certain a.s.signment of value to it gave, as we shall see, Thomson's famous absolute thermodynamic scale of temperature.

In the light of the facts and theories which now exist, and are almost the commonplaces of physical text books, it is very interesting to review the ideas and difficulties which occurred to the founders of the science of heat sixty years ago. For example, Thomson asks, in his "Account of Carnot's Theory," what becomes of the mechanical effect which might be produced by heat which is transferred from one body to another by conduction. The heat leaves one body and enters another and no mechanical effect results: if it pa.s.sed from one to the other through a heat engine, mechanical effect would be produced: what is produced in place of the mechanical effect which is lost? This he calls a very "perplexing question," and hopes that it will, before long, be cleared up. He states, further, that the difficulty would be entirely avoided by abandoning Carnot's principle that mechanical effect is obtained by "the transference of heat from one body to another at a lower temperate."

Joule urges precisely this solution of the difficulty in his paper, "On the Changes of Temperature produced by the Rarefaction and Condensation of Air" (_Phil. Mag._, May 1845). Thomson notes this, but adds, "If we do so, however, we meet with innumerable other difficulties--insuperable without further experimental investigation, and an entire reconstruction of the theory of heat from its foundation. It is in reality to experiment that we must look, either for a verification of Carnot's axiom, and an explanation of the difficulty we have been considering, or for an entirely new basis of the Theory of Heat."

The experiments here asked for had already, as was soon after perceived by Thomson, been made by Joule, not merely in his determinations of the dynamical equivalent of heat, but in his exceedingly important investigation of the energy changes in the circuit of a voltaic cell, or of a magneto-electric machine. Moreover, the answer to this "very perplexing question" was afterwards to be given by Thomson himself in his paper, "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy," published in the Edinburgh Proceedings in 1852.

Again, we find, a page or two earlier in the "Account of Carnot's Theory," the question asked with respect to the heat evolved in the circuit of a magneto-electric machine, "Is the heat which is evolved in one part of the closed conductor merely transferred from those parts which are subject to the inducing influence?" and the statement made that Joule had examined this question, and decided that it must be answered in the negative. But Thomson goes on to say, "Before we can finally conclude that heat is absolutely generated in such operations, it would be necessary to prove that the inducing magnet does not become lower in temperature and thus compensate for the heat evolved in the conductor."

Here, apparently, the idea of work done in moving the magnet, or the conductor in the magnetic field, is not present to Thomson's mind; for if it had been, the idea that the work thus spent might have its equivalent, in part, at least, in heat generated in the circuit, would no doubt have occurred to him and been stated. This idea had been used just a year before by Helmholtz, in his essay "Die Erhaltung der Kraft,"

to account for the heat produced in the circuit by the induced current, that is, to answer the first question put above in the sense in which Joule answered it. The subject, however, was fully worked out by Thomson in a paper published in the _Philosophical Magazine_ for December 1851, to which we shall refer later.

Tables of the work performed by various steam engines working between different stated temperatures were given at the close of the "Account,"

and compared with the theoretical "duty" as calculated for Carnot's ideal perfect engine. Of course the theoretical duty was calculated from the temperatures of the boiler and condenser; the much greater fall of temperature from the furnace to the boiler was neglected as inevitable, so that the loss involved in that fall is not taken account of. Carnot's theory gave for the theoretical duty of one heat unit (equivalent to 1390 foot-pounds of work) 440 foot-pounds for boiler at 140 C. and condenser at 30 C.; and the best performance recorded was 253 foot-pounds, giving a percentage of 57.5 per cent. The worst was that of common engines consuming 12 lb. of coal per horse-power per hour, and gave 38.1 foot-pounds, or a percentage of 8.6 per cent. These percentages become on the dynamical theory 68 and 10.3, since the true theoretical duty for the heat unit is only 371 foot-pounds.

It is worthy of notice that the indicator-diagram method of graphically representing the changes in a cycle of operations is adopted in Thomson's "Account," but does not occur in Carnot's essay. The cycles consist of two isothermal changes and two adiabatic changes; that is, two changes at the temperatures of the source and refrigerator respectively, and two changes--from the higher to the lower temperature, and from the lower to the higher. These changes are made subject to the condition in each case that the substance neither gains nor loses energy in the form of heat, but is cooled in the one case by expansion and heated in the other by compression. The indicator diagram was due not to Thomson but to Clapeyron (see p. 99 above), who used it to ill.u.s.trate an account of Carnot's theory.

There appeared in the issue of the Edinburgh _Philosophical Transactions_ for January 2, 1849, along with the "Account of Carnot's Theory," a paper by James Thomson, ent.i.tled, "Theoretical Considerations on the Effect of Pressure in Lowering the Freezing Point of Water." The author predicted that, unless the principle of conservation of energy was at fault, the effect of increase of pressure on water in the act of freezing would be to lower the freezing point; and he calculated from Carnot's theory the amount of lowering which would be produced by a given increment of pressure. The prediction thus made was tested by experiments carried out in the Physical Laboratory by Thomson, and the results obtained completely confirmed the conclusions arrived at by theory. This prediction and its verification have been justly regarded as of great importance in the history of the dynamical theory of heat; and they afford an excellent example of the predictive character of a true scientific theory. The theory of the matter will be referred to in the next chapter.